We investigate a robust optimal reinsurance-investment problem for $ n $ insurers under multiple interactions, which arise from the insurance market, the financial market, the competition mechanism and the cooperation mechanism. Each insurer's surplus process is assumed to follow a diffusion model, which is an approximation of the classical Cramér-Lundberg model. Each insurer is allowed to purchase proportional reinsurance to reduce their claim risk. To reflect the first moment and second moment information on claims, we use the variance premium principle to calculate reinsurance premiums. To increase wealth, each insurer can invest in a financial market, which includes one risk-free asset and $ n $ correlated stocks. Each insurer wants to obtain the robust optimal reinsurance and investment strategy under the mean-variance criterion. By applying a stochastic control technique and dynamic programming approach, the extended Hamilton-Jacobi-Bellman (HJB) equation is established. Furthermore, we derive both the robust optimal reinsurance-investment strategy and the corresponding value function by solving the extended HJB equation. Finally, we present numerical experiments, which yield that competition and cooperation have an important influence on the insurer's decision-making.
Citation: Peng Yang. Robust optimal reinsurance-investment problem for $ n $ competitive and cooperative insurers under ambiguity aversion[J]. AIMS Mathematics, 2023, 8(10): 25131-25163. doi: 10.3934/math.20231283
We investigate a robust optimal reinsurance-investment problem for $ n $ insurers under multiple interactions, which arise from the insurance market, the financial market, the competition mechanism and the cooperation mechanism. Each insurer's surplus process is assumed to follow a diffusion model, which is an approximation of the classical Cramér-Lundberg model. Each insurer is allowed to purchase proportional reinsurance to reduce their claim risk. To reflect the first moment and second moment information on claims, we use the variance premium principle to calculate reinsurance premiums. To increase wealth, each insurer can invest in a financial market, which includes one risk-free asset and $ n $ correlated stocks. Each insurer wants to obtain the robust optimal reinsurance and investment strategy under the mean-variance criterion. By applying a stochastic control technique and dynamic programming approach, the extended Hamilton-Jacobi-Bellman (HJB) equation is established. Furthermore, we derive both the robust optimal reinsurance-investment strategy and the corresponding value function by solving the extended HJB equation. Finally, we present numerical experiments, which yield that competition and cooperation have an important influence on the insurer's decision-making.
| [1] | T. Björk, A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, Working Paper, Stockholm School of Economics, 2010. https://doi.org/10.2139/ssrn.1694759 |
| [2] |
S. Basak, G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Financ. Stud., 23 (2010), 2970–3016. https://doi.org/10.1093/rfs/hhq028 doi: 10.1093/rfs/hhq028
|
| [3] |
P. Yang, Time-consistent mean-variance reinsurance-investment in a jump-diffusion financial market, Optimization, 66 (2017), 737–758. https://doi.org/10.1080/02331934.2017.1296837 doi: 10.1080/02331934.2017.1296837
|
| [4] |
P. Yang, Z. Chen, L. Wang, Time-consistent reinsurance and investment strategy combining quota-share and excess of loss for meanvariance insurers with jump-diffusion price process, Commun. Stat. Theor. M., 50 (2021), 2546–2568. https://doi.org/10.1080/03610926.2019.1670849 doi: 10.1080/03610926.2019.1670849
|
| [5] |
Y. Wang, Y. Deng, Y. Huang, J. Zhou, X. Xiang, Optimal reinsurance-investment policies for insurers with mispricing under mean-variance criterion, Commun. Stat. Theor. M., 51 (2022), 5653–5680. https://doi.org/10.1080/03610926.2020.1844239 doi: 10.1080/03610926.2020.1844239
|
| [6] | C. Zhang, Z. Liang, Optimal time-consistent reinsurance and investment strategies for a jump-diffusion financial market without cash, N. Am. J. Econ. Financ., 59 (2022), Article ID 101578. https://doi.org/10.1016/j.najef.2021.101578 |
| [7] |
L. Wang, M. C. Chiu, H. Y. Wong, Time-consistent mean-variance reinsurance-investment problem with long-range dependent mortality rate, Scand. Actuar. J., 2023 (2023b), 123–152. https://doi.org/10.1080/03461238.2022.2089050 doi: 10.1080/03461238.2022.2089050
|
| [8] |
A. Bensoussan, C. C. Siu, S. C. P. Yam, H. L. Yang, A class of non-zero-sum stochastic diffrential investment and reinsurance games, Automatica, 50 (2014), 2025–2037. https://doi.org/10.1016/j.automatica.2014.05.033 doi: 10.1016/j.automatica.2014.05.033
|
| [9] |
C. C. Siu, S. C. P. Yam, H. Yang, H. Zhao, A class of nonzero-sum investment and reinsurance games subject to systematic risks, Scand. Actuar. J., 2017 (2017), 670–707. https://doi.org/10.1080/03461238.2016.1228542 doi: 10.1080/03461238.2016.1228542
|
| [10] |
C. Deng, X. Zeng, H. Zhu, Non-zero-sum stochastic differential reinsurance and investment games with default risk, Eur. J. Oper. Res., 264 (2018), 1144–1158. https://doi.org/10.1016/j.ejor.2017.06.065 doi: 10.1016/j.ejor.2017.06.065
|
| [11] |
D. Hu, H. Wang, Time-consistent investment and reinsurance under relative performance concerns, Commun. Stat. Theor. M., 47 (2018), 1693–1717. https://doi.org/10.1080/03610926.2017.1324987 doi: 10.1080/03610926.2017.1324987
|
| [12] |
H. Zhu, M. Cao, C. K. Zhang, Time-consistent investment and reinsurance strategies for mean-variance insurers with relative performance concerns under the Heston model, Financ. Res. Lett., 30 (2019), 280–291. https://doi.org/10.1016/j.frl.2018.10.009 doi: 10.1016/j.frl.2018.10.009
|
| [13] |
Y. Bai, Z. Zhou, H. Xiao, R. Gao, F. Zhong, A hybrid stochastic differential reinsurance and investment game with bounded memory, Eur. J. Oper. Res., 296 (2022), 717–737. https://doi.org/10.1016/j.ejor.2021.04.046 doi: 10.1016/j.ejor.2021.04.046
|
| [14] |
X. Dong, X. Rong, H. Zhao, Non-zero-sum reinsurance and investment game with non-trivial curved strategy structure under Ornstein-Uhlenbeck process, Scand. Actuar. J., 2023 (2023), 565–597. https://doi.org/10.1080/03461238.2022.2139631 doi: 10.1080/03461238.2022.2139631
|
| [15] | P. Yang, Z. Chen, Y. Xu, Time-consistent equilibrium reinsurance-investment strategy for n competitive insurers under a new interaction mechanism and a general investment framework, J. Comput. Appl. Math., 374 (2020), Article ID 112769. https://doi.org/10.1016/j.cam.2020.112769 |
| [16] |
G. Guan, X. Hu, Time-consistent investment and reinsurance strategies for mean-variance insurers in $N$-agent and mean-field games, N. Am. Actuar J., 26 (2022), 537–569. https://doi.org/10.1080/10920277.2021.2014891 doi: 10.1080/10920277.2021.2014891
|
| [17] |
E. W. Anderson, L. P. Hansen, T. J. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, J. Eur. Econ. Assoc., 1 (2003), 68–123. https://doi.org/10.1162/154247603322256774 doi: 10.1162/154247603322256774
|
| [18] |
R. Uppal, T. Wang, Model misspecification and underdiversification, J. Finance, 58 (2003), 2465–2486. https://doi.org/10.1046/j.1540-6261.2003.00612.x doi: 10.1046/j.1540-6261.2003.00612.x
|
| [19] |
P. J. Maenhout, Robust portfolio rules and asset pricing, Rev. Financ. Stud., 17 (2004), 951–983. https://doi.org/10.1093/rfs/hhh003 doi: 10.1093/rfs/hhh003
|
| [20] |
B. Li, D. Li, D. Xiong, Alpha-robust mean-variance reinsurance-investment strategy, J. Econ. Dyn. Control., 70 (2016), 101–123. https://doi.org/10.1016/j.jedc.2016.07.001 doi: 10.1016/j.jedc.2016.07.001
|
| [21] |
Y. Zeng, D. Li, A. Gu, Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insur. Math. Econ., 66 (2016), 138–152. https://doi.org/10.1016/j.insmatheco.2015.10.012 doi: 10.1016/j.insmatheco.2015.10.012
|
| [22] |
D. Li, Y. Zeng, H. Yao, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scand. Actuar. J., 2017 (2017), 1–27. https://doi.org/10.1080/03461238.2017.1309679 doi: 10.1080/03461238.2017.1309679
|
| [23] |
C. C. Pun, Robust time-inconsistent stochastic control problems, Automatica, 94 (2018), 249–257. https://doi.org/10.1016/j.automatica.2018.04.038 doi: 10.1016/j.automatica.2018.04.038
|
| [24] |
Z. Chen, P. Yang, Robust optimal reinsurance-investment strategy with price jumps and correlated claims, Insur. Math. Econ., 92 (2020), 27–46. https://doi.org/10.1016/j.insmatheco.2020.03.001 doi: 10.1016/j.insmatheco.2020.03.001
|
| [25] |
X. Peng, W. Wang, Optimal investment and risk control for an insurer under inside information, Insur. Math. Econ., 69 (2016), 104–116. https://doi.org/10.1016/j.insmatheco.2016.04.008 doi: 10.1016/j.insmatheco.2016.04.008
|
| [26] | P. Yang, Closed-loop equilibrium reinsurance-investment strategy with insider information and default risk, Math. Probl. Eng., 2021 (2021), Article ID 8873473. https://doi.org/10.1155/2021/8873473 |
| [27] |
X. Peng, F. Chen, W. Wang, Robust optimal investment and reinsurance for an insurer with inside information, Insur. Math. Econ., 96 (2021), 15–30. https://doi.org/10.1016/j.insmatheco.2020.10.004 doi: 10.1016/j.insmatheco.2020.10.004
|
| [28] |
W. Jiang, Z. Yang, Optimal robust insurance contracts with investment strategy under variance premium principle, Math. Control Relat. F., (2023). https://doi.org/10.3934/mcrf.2023001 doi: 10.3934/mcrf.2023001
|
| [29] |
N. Wang, N. Zhang, Z. Jin, L. Qian, Robust non-zero-sum investment and reinsurance game with default risk, Insur. Math. Econ., 84 (2019), 115–132. https://doi.org/10.1016/j.insmatheco.2018.09.009 doi: 10.1016/j.insmatheco.2018.09.009
|
| [30] |
P. Yang, Robust optimal reinsurance strategy with correlated claims and competition, AIMS Math., 8 (2023), 15689–15711. https://doi.org/10.3934/math.2023801 doi: 10.3934/math.2023801
|
| [31] | I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, Springer-Verlag, New York, 1988. https://doi.org/10.1007/978-1-4684-0302-2 |
| [32] |
Z. Chen, P. Yang, Y. Gan, Optimal reinsurance and investment with a common shock and a random exit time, Rairo-Oper. Res., 57 (2023), 881–903. https://doi.org/10.1051/ro/2023036 doi: 10.1051/ro/2023036
|
| [33] |
L. Xu, L. Wang, X. Liu, H. Wang, Optimal active lifetime investment, Int. J. Control, 96 (2023), 48–57. https://doi.org/10.1080/00207179.2021.1979252 doi: 10.1080/00207179.2021.1979252
|
| [34] | J. Grandell, Aspects of risk theory, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4613-9058-9 |
| [35] |
S. Li, W. Yuan, P. Chen, Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market, J. Ind. Manag. Optim., (2022). https://doi.org/10.3934/jimo.2022068 doi: 10.3934/jimo.2022068
|
| [36] |
J. Ma, H. Zha, X. Rong, Optimal investment strategy for a DC pension plan with mispricing under the Heston model, Commun. Stat.-Theor. M., 49 (2020), 3168–3183. https://doi.org/10.1080/03610926.2019.1586938 doi: 10.1080/03610926.2019.1586938
|
| [37] |
Z. Liu, Y. Wang, Y. Huang, J. Zhou, Optimal portfolios for the DC pension fund with mispricing under the HARA utility framework, J. Ind. Manag. Optim., 19 (2023), 1262–1281. https://doi.org/10.3934/jimo.2021228 doi: 10.3934/jimo.2021228
|
| [38] | J. Kang, M. Wang, N. Huang, Equilibrium strategy for mean-variance-utility portfolio selection under Heston's SV model, J. Comput. Appl. Math., 392 (2021), Article ID 113490. https://doi.org/10.1016/j.cam.2021.113490 |
| [39] |
W. Wang, D. Muravey, Y. Shen, Y. Zeng, Optimal investment and reinsurance strategies under $4/2$ stochastic volatility mode, Scand. Actuar. J., 2023 (2023a), 413–449. https://doi.org/10.1080/03461238.2022.2108335 doi: 10.1080/03461238.2022.2108335
|
| [40] |
L. Xu, S. Xu, D. Yao, Maximizing expected terminal utility of an insurer with high gain tax by investment and reinsurance, Comput. Math. Appl., 79 (2020), 716–734. https://doi.org/10.1016/j.camwa.2019.07.023 doi: 10.1016/j.camwa.2019.07.023
|
| [41] |
G. E. Espinosa, N. Touzi, Optimal investment under relative performance concerns, Math. Finace., 25 (2015), 221–257. https://doi.org/10.1111/mafi.12034 doi: 10.1111/mafi.12034
|
| [42] |
T. Björk, A. Murgoci, X. Y. Zhou, Mean-variance portfolio optimization with state dependent risk aversion, Math. Finance, 24 (2014), 1–24. https://doi.org/10.1111/j.1467-9965.2011.00515.x doi: 10.1111/j.1467-9965.2011.00515.x
|
| [43] |
B. Yi, Z. Li, F. G. Viens, Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under Heston's stochastic volatility model, Insur. Math. Econ., 53 (2013), 601–614. https://doi.org/10.1016/j.insmatheco.2013.08.011 doi: 10.1016/j.insmatheco.2013.08.011
|
| [44] |
Y. Zeng, Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insur. Math. Econ., 49 (2011), 145–154. https://doi.org/10.1016/j.insmatheco.2011.01.001 doi: 10.1016/j.insmatheco.2011.01.001
|
| [45] |
J. Zhou, X. Yang, Y. Huang, Robust optimal investment and proportional reinsurance toward joint interests of the insurer and the reinsurer, Commun. Stat. Theor. M., 46 (2017), 10733–10757. https://doi.org/10.1080/03610926.2016.1242734 doi: 10.1080/03610926.2016.1242734
|
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Peng Yang. Robust optimal reinsurance-investment problem for $ n $ competitive and cooperative insurers under ambiguity aversion[J]. AIMS Mathematics, 2023, 8(10): 25131-25163. doi: 10.3934/math.20231283
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